cern.colt

Class Partitioning

java.lang.Object

cern.colt.Partitioning

public class Partitioning
extends Object

Given some interval boundaries, partitions arrays such that all elements falling into an interval are placed next to each other.

The algorithms partition arrays into two or more intervals. They distinguish between synchronously partitioning either one, two or three arrays. They further come in templated versions, either partitioning int[] arrays or double[] arrays.

You may want to start out reading about the simplest case: Partitioning one int[] array into two intervals. To do so, read partition(int[],int,int,int). Next, building upon that foundation comes a method partitioning int[] arrays into multiple intervals. See partition(int[],int,int,int[],int,int,int[]) for related documentation.

All other methods are no different than the one's you now already understand, except that they operate on slightly different data types.

Performance

Partitioning into two intervals is O( N ). Partitioning into k intervals is O( N * log(k)). Constants factors are minimized. No temporary memory is allocated; Partitioning is in-place.

Version:

1.0, 03-Jul-99

Author:

wolfgang.hoschek@cern.ch

See Also:

Partitioning

Field Summary

Fields

Modifier and Type	Field and Description
protected static int	steps
static int	swappedElements

Constructor Summary

Constructors

Modifier	Constructor and Description
protected	Partitioning() Makes this class non instantiable, but still let's others inherit from it.

Method Summary

Methods

Modifier and Type	Method and Description
static int	dualPartition(double[] list, double[] secondary, int from, int to, double splitter) Same as dualPartition(int[],int[],int,int,int) except that it synchronously partitions double[] rather than int[] arrays.
static void	dualPartition(double[] list, double[] secondary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same as dualPartition(int[],int[],int,int,int[],int,int,int[]) except that it synchronously partitions double[] rather than int[] arrays.
static int	dualPartition(int[] list, int[] secondary, int from, int to, int splitter) Same as partition(int[],int,int,int) except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array.
static void	dualPartition(int[] list, int[] secondary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same as partition(int[],int,int,int[],int,int,int[]) except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array.
static void	genericPartition(int from, int to, int splitFrom, int splitTo, int[] splitIndexes, IntComparator comp, IntComparator comp2, IntComparator comp3, Swapper swapper) Same as partition(int[],int,int,int[],int,int,int[]) except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays.
static int	partition(double[] list, int from, int to, double splitter) Same as partition(int[],int,int,int) except that it partitions double[] rather than int[] arrays.
static void	partition(double[] list, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same as partition(int[],int,int,int[],int,int,int[]) except that it partitions double[] rather than int[] arrays.
static void	partition(DoubleArrayList list, int from, int to, DoubleArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).
static int	partition(int[] list, int from, int to, int splitter) Partitions (partially sorts) the given list such that all elements falling into the given interval are placed next to each other.
static void	partition(int[] list, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Partitions (partially sorts) the given list such that all elements falling into some intervals are placed next to each other.
static void	partition(IntArrayList list, int from, int to, IntArrayList splitters, IntArrayList splitIndexes) Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).
static void	partition(Object[] list, int from, int to, Object[] splitters, int splitFrom, int splitTo, int[] splitIndexes, Comparator comp) Same as partition(int[],int,int,int[],int,int,int[]) except that it partitions Object[] rather than int[] arrays.
static int	partition(Object[] list, int from, int to, Object splitter, Comparator comp) Same as partition(int[],int,int,int) except that it synchronously partitions the objects of the given list by the order of the given comparator.
static int	triplePartition(double[] list, double[] secondary, double[] tertiary, int from, int to, double splitter) Same as triplePartition(int[],int[],int[],int,int,int) except that it synchronously partitions double[] rather than int[] arrays.
static void	triplePartition(double[] list, double[] secondary, double[] tertiary, int from, int to, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same as triplePartition(int[],int[],int[],int,int,int[],int,int,int[]) except that it synchronously partitions double[] rather than int[] arrays.
static int	triplePartition(int[] list, int[] secondary, int[] tertiary, int from, int to, int splitter) Same as partition(int[],int,int,int) except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array.
static void	triplePartition(int[] list, int[] secondary, int[] tertiary, int from, int to, int[] splitters, int splitFrom, int splitTo, int[] splitIndexes) Same as partition(int[],int,int,int[],int,int,int[]) except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

steps

protected static int steps

swappedElements

public static int swappedElements

Constructor Detail

Partitioning

protected Partitioning()

Makes this class non instantiable, but still let's others inherit from it.

Method Detail

dualPartition

public static void dualPartition(double[] list,
                 double[] secondary,
                 int from,
                 int to,
                 double[] splitters,
                 int splitFrom,
                 int splitTo,
                 int[] splitIndexes)

Same as dualPartition(int[],int[],int,int,int[],int,int,int[]) except that it synchronously partitions double[] rather than int[] arrays.

dualPartition

public static int dualPartition(double[] list,
                double[] secondary,
                int from,
                int to,
                double splitter)

Same as dualPartition(int[],int[],int,int,int) except that it synchronously partitions double[] rather than int[] arrays.

dualPartition

public static void dualPartition(int[] list,
                 int[] secondary,
                 int from,
                 int to,
                 int[] splitters,
                 int splitFrom,
                 int splitTo,
                 int[] splitIndexes)

Same as partition(int[],int,int,int[],int,int,int[]) except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array is also moved from index A to B.

Use cases:

Image having a large list of 2-dimensional points. If memory consumption and performance matter, it is a good idea to physically lay them out as two 1-dimensional arrays (using something like Point2D objects would be prohibitively expensive, both in terms of time and space). Now imagine wanting to histogram the points. We may want to partially sort the points by x-coordinate into intervals. This method efficiently does the job.

Performance:

Same as for single-partition methods.

dualPartition

public static int dualPartition(int[] list,
                int[] secondary,
                int from,
                int to,
                int splitter)

Same as partition(int[],int,int,int) except that this method synchronously partitions two arrays at the same time; both arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array is also moved from index A to B.

Performance:

Same as for single-partition methods.

genericPartition

public static void genericPartition(int from,
                    int to,
                    int splitFrom,
                    int splitTo,
                    int[] splitIndexes,
                    IntComparator comp,
                    IntComparator comp2,
                    IntComparator comp3,
                    Swapper swapper)

Same as partition(int[],int,int,int[],int,int,int[]) except that it generically partitions arbitrary shaped data (for example matrices or multiple arrays) rather than int[] arrays.

This method operates on arbitrary shaped data and arbitrary shaped splitters. In fact, it has no idea what kind of data by what kind of splitters it is partitioning. Comparisons and swapping are delegated to user provided objects which know their data and can do the job.

Lets call the generic data g (it may be a matrix, one array, three linked lists or whatever). Lets call the generic splitters s. This class takes a user comparison function operating on two indexes (a,b), namely an IntComparator. The comparison function determines whether s[a] is equal, less or greater than g[b]. This method can then decide to swap the data g[b] with the data g[c] (yes, c, not a). It calls a user provided Swapper object that knows how to swap the data of these two indexes.

Again, note the details: Comparisons compare s[a] with g[b]. Swaps swap g[b] with g[c]. Prior to calling this method, the generic splitters s must be sorted ascending and must not contain multiple equal values. These preconditions are not checked; be sure that they are met.

Parameters:

from - the index of the first element within g to be considered.

to - the index of the last element within g to be considered. The method considers the elements g[from] .. g[to].

splitFrom - the index of the first splitter element to be considered.

splitTo - the index of the last splitter element to be considered. The method considers the splitter elements s[splitFrom] .. s[splitTo].

splitIndexes - a list into which this method fills the indexes of elements delimiting intervals. Upon return splitIndexes[splitFrom..splitTo] will be set accordingly. Therefore, must satisfy splitIndexes.length > splitTo.

comp - the comparator comparing a splitter with an element of the generic data. Takes as first argument the index a within the generic splitters s. Takes as second argument the index b within the generic data g.

comp2 - the comparator to determine the order of the generic data. Takes as first argument the index a within the generic data g. Takes as second argument the index b within the generic data g.

comp3 - the comparator comparing a splitter with another splitter. Takes as first argument the index a within the generic splitters s. Takes as second argument the index b within the generic splitters g.

swapper - an object that knows how to swap the elements at any two indexes (a,b). Takes as first argument the index b within the generic data g. Takes as second argument the index c within the generic data g.

Tip: Normally you will have splitIndexes.length == s.length as well as from==0, to==g.length-1 and splitFrom==0, splitTo==s.length-1.

See Also:

Sort, Sort#sort(int,int,IntComparator,Swapper), Sorting.binarySearchFromTo(int,int,IntComparator)

partition

public static void partition(double[] list,
             int from,
             int to,
             double[] splitters,
             int splitFrom,
             int splitTo,
             int[] splitIndexes)

Same as partition(int[],int,int,int[],int,int,int[]) except that it partitions double[] rather than int[] arrays.

partition

public static int partition(double[] list,
            int from,
            int to,
            double splitter)

Same as partition(int[],int,int,int) except that it partitions double[] rather than int[] arrays.

partition

public static void partition(int[] list,
             int from,
             int to,
             int[] splitters,
             int splitFrom,
             int splitTo,
             int[] splitIndexes)

Partitions (partially sorts) the given list such that all elements falling into some intervals are placed next to each other. Returns the indexes of elements delimiting intervals.

Example:

list = (7, 4, 5, 50, 6, 4, 3, 6), splitters = (5, 10, 30) defines the three intervals [-infinity,5), [5,10), [10,30). Lets define to sort the entire list (from=0, to=7) using all splitters (splitFrom==0, splitTo=2).

The method modifies the list to be list = (4, 4, 3, 6, 7, 5, 6, 50) and returns the splitIndexes = (2, 6, 6). In other words,

• All values list[0..2] fall into [-infinity,5).
• All values list[3..6] fall into [5,10).
• All values list[7..6] fall into [10,30), i.e. no elements, since 7>6.
• All values list[7 .. 7=list.length-1] fall into [30,infinity].
• In general, all values list[splitIndexes[j-1]+1 .. splitIndexes[j]] fall into interval j.

As can be seen, the list is partially sorted such that values falling into a certain interval are placed next to each other. Note that within an interval, elements are entirelly unsorted. They are only sorted across interval boundaries. In particular, this partitioning algorithm is not stable: the relative order of elements is not preserved (Producing a stable algorithm would require no more than minor modifications to method partition(int[],int,int,int)).

More formally, this method guarantees that upon return for all j = splitFrom .. splitTo there holds:
for all i = splitIndexes[j-1]+1 .. splitIndexes[j]: splitters[j-1] <= list[i] < splitters[j].

Performance:

Let N=to-from+1 be the number of elements to be partitioned. Let k=splitTo-splitFrom+1 be the number of splitter elements. Then we have the following time complexities

• Worst case: O( N * log(k) ).
• Average case: O( N * log(k) ).
• Best case: O( N ). In general, the more uniform (skewed) the data is spread across intervals, the more performance approaches the worst (best) case. If no elements fall into the given intervals, running time is linear.

No temporary memory is allocated; the sort is in-place.

Implementation:

The algorithm can be seen as a Bentley/McIlroy quicksort where swapping and insertion sort are omitted. It is designed to detect and take advantage of skew while maintaining good performance in the uniform case.

Parameters:

list - the list to be partially sorted.

from - the index of the first element within list to be considered.

to - the index of the last element within list to be considered. The method considers the elements list[from] .. list[to].

splitters - the values at which the list shall be split into intervals. Must be sorted ascending and must not contain multiple identical values. These preconditions are not checked; be sure that they are met.

splitFrom - the index of the first splitter element to be considered.

splitTo - the index of the last splitter element to be considered. The method considers the splitter elements splitters[splitFrom] .. splitters[splitTo].

splitIndexes - a list into which this method fills the indexes of elements delimiting intervals. Upon return splitIndexes[splitFrom..splitTo] will be set accordingly. Therefore, must satisfy splitIndexes.length > splitTo.

Tip: Normally you will have splitIndexes.length == splitters.length as well as from==0, to==list.length-1 and splitFrom==0, splitTo==splitters.length-1.

See Also:

Arrays, GenericSorting, Arrays

partition

public static int partition(int[] list,
            int from,
            int to,
            int splitter)

Partitions (partially sorts) the given list such that all elements falling into the given interval are placed next to each other. Returns the index of the element delimiting the interval.

Example:

list = (7, 4, 5, 50, 6, 4, 3, 6), splitter = 5 defines the two intervals [-infinity,5), [5,+infinity].

The method modifies the list to be list = (4, 4, 3, 50, 6, 7, 5, 6) and returns the split index 2. In other words,

• All values list[0..2] fall into [-infinity,5).
• All values list[3=2+1 .. 7=list.length-1] fall into [5,+infinity].

As can be seen, the list is partially sorted such that values falling into a certain interval are placed next to each other. Note that within an interval, elements are entirelly unsorted. They are only sorted across interval boundaries. In particular, this partitioning algorithm is not stable.

More formally, this method guarantees that upon return there holds:

• for all i = from .. returnValue: list[i] < splitter and
• for all i = returnValue+1 .. list.length-1: !(list[i] < splitter).

Performance:

Let N=to-from+1 be the number of elements to be partially sorted. Then the time complexity is O( N ). No temporary memory is allocated; the sort is in-place.

Parameters:

list - the list to be partially sorted.

from - the index of the first element within list to be considered.

to - the index of the last element within list to be considered. The method considers the elements list[from] .. list[to].

splitter - the value at which the list shall be split.

Returns:

the index of the largest element falling into the interval [-infinity,splitter), as seen after partitioning.

partition

public static void partition(Object[] list,
             int from,
             int to,
             Object[] splitters,
             int splitFrom,
             int splitTo,
             int[] splitIndexes,
             Comparator comp)

Same as partition(int[],int,int,int[],int,int,int[]) except that it partitions Object[] rather than int[] arrays.

partition

public static int partition(Object[] list,
            int from,
            int to,
            Object splitter,
            Comparator comp)

Same as partition(int[],int,int,int) except that it synchronously partitions the objects of the given list by the order of the given comparator.

partition

public static void partition(DoubleArrayList list,
             int from,
             int to,
             DoubleArrayList splitters,
             IntArrayList splitIndexes)

Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).

partition

public static void partition(IntArrayList list,
             int from,
             int to,
             IntArrayList splitters,
             IntArrayList splitIndexes)

Equivalent to partition(list.elements(), from, to, splitters.elements(), 0, splitters.size()-1, splitIndexes.elements()).

triplePartition

public static void triplePartition(double[] list,
                   double[] secondary,
                   double[] tertiary,
                   int from,
                   int to,
                   double[] splitters,
                   int splitFrom,
                   int splitTo,
                   int[] splitIndexes)

Same as triplePartition(int[],int[],int[],int,int,int[],int,int,int[]) except that it synchronously partitions double[] rather than int[] arrays.

triplePartition

public static int triplePartition(double[] list,
                  double[] secondary,
                  double[] tertiary,
                  int from,
                  int to,
                  double splitter)

Same as triplePartition(int[],int[],int[],int,int,int) except that it synchronously partitions double[] rather than int[] arrays.

triplePartition

public static void triplePartition(int[] list,
                   int[] secondary,
                   int[] tertiary,
                   int from,
                   int to,
                   int[] splitters,
                   int splitFrom,
                   int splitTo,
                   int[] splitIndexes)

Same as partition(int[],int,int,int[],int,int,int[]) except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array as well as the corresponding element within the tertiary array are also moved from index A to B.

Use cases:

Image having a large list of 3-dimensional points. If memory consumption and performance matter, it is a good idea to physically lay them out as three 1-dimensional arrays (using something like Point3D objects would be prohibitively expensive, both in terms of time and space). Now imagine wanting to histogram the points. We may want to partially sort the points by x-coordinate into intervals. This method efficiently does the job.

Performance:

Same as for single-partition methods.

triplePartition

public static int triplePartition(int[] list,
                  int[] secondary,
                  int[] tertiary,
                  int from,
                  int to,
                  int splitter)

Same as partition(int[],int,int,int) except that this method synchronously partitions three arrays at the same time; all three arrays are partially sorted according to the elements of the primary array. In other words, each time an element in the primary array is moved from index A to B, the correspoding element within the secondary array as well as the corresponding element within the tertiary array are also moved from index A to B.

Performance:

Same as for single-partition methods.